Primitive digraphs with large exponents and slowly synchronizing automata

Ananichev, D. S.; Volkov, M. V.; Gusev, Vladimir V.

doi:10.1007/s10958-013-1392-8

Cited by 50 publications

(108 citation statements)

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“…As mentioned, the proof for the slowly synchronizing automata defined in (Ananichev et al 2012) is very similar. The most important difference is the growth factor in BFS steps, but it is exponential in all the cases.…”

Section: Theorem 3 For the Class Of Thečerný Automata C N [And All mentioning

confidence: 64%

“…In (Ananichev et al 2012), the authors introduce the series of, what they call, slowly synchronizing automata D n ,W n ,F n ,E n ,D n ,B n ,G n ,H n with the property that the reset length of these automata is quadratic in terms of the number of states n and close to theČerný bound (n − 1) 2 . Now, while, generally, our algorithm is exponential in the reset length l, surprisingly, it works fast in polynomial time for all the slowly synchronizing automata defined in (Ananichev et al 2012). Since the proof is different but very similar in each case, we demonstrate it only for theČerný automata C n .…”

Section: Performance On Slowly Synchronizing Automatamentioning

confidence: 99%

“…Since the proof is different but very similar in each case, we demonstrate it only for theČerný automata C n . In other cases the proof is left to the reader on the basis of the definitions given in (Ananichev et al 2012). (Ananichev et al 2012)] the algorithm works in O(n 4 ) time and O(n 2 log n) space.…”

Section: Performance On Slowly Synchronizing Automatamentioning

confidence: 99%

“…So far, the conjecture has been proved only for some special classes of automata and a general cubic upper bound (n 3 − n)/6 has been established (see Volkov (2008) for an excellent survey of the results). Using computers the conjecture has been verified for small automata with 2 letters and n ≤ 11 states (Kisielewicz and Szykuła 2013) (and with k ≤ 4 letters and n ≤ 7 states (Trahtman 2006); see also (Ananichev et al 2010(Ananichev et al , 2012 for n = 9 states). It is known that, in general, the problem is computationally hard, since it involves an NP-hard decision problem.…”

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confidence: 94%

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Computing the shortest reset words of synchronizing automata

2013

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In this paper we give the details of our new algorithm for finding minimal reset words of finite synchronizing automata. The problem is known to be computationally hard, so our algorithm is exponential in the worst case, but it is faster than the algorithms used so far and it performs well on average. The main idea is to use a bidirectional breadth-first-search and radix (Patricia) tries to store and compare subsets. A good performance is due to a number of heuristics we apply and describe here in a suitable detail. We give both theoretical and practical arguments showing that the effective branching factor is considerably reduced. As a practical test we perform an experimental study of the length of the shortest reset word for random automata with up to n = 350 states and up to k = 10 input letters. In particular, we obtain a new estimation of the expected length of the shortest reset word ≈ 2.5 √ n − 5 for binary automata and show that the error of this estimate is sufficiently small. Experiments for automata with more than two input letters show certain trends with the same general pattern.

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Section: Theorem 3 For the Class Of Thečerný Automata C N [And All mentioning

confidence: 64%

Section: Performance On Slowly Synchronizing Automatamentioning

confidence: 99%

Section: Performance On Slowly Synchronizing Automatamentioning

confidence: 99%

mentioning

confidence: 94%

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Computing the shortest reset words of synchronizing automata

2013

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“…This observation may be immediately applied to prove the simplicity of other slowly synchronizing automata, like the Wielandt automaton W n , and the automaton D n described in [4] (see Fig. 1).…”

Section: Semisimple Synchronizing Automata and Radical Wordsmentioning

confidence: 90%

Semisimple Synchronizing Automata and the Wedderburn-Artin Theory

Almeida

Rodaro

2016

Int. J. Found. Comput. Sci.

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Abstract. We present a ring theoretic approach toČerný's conjecture via the Wedderburn-Artin theory. We first introduce the radical ideal of a synchronizing automaton, and then the natural notion of semisimple synchronizing automata. This is a rather broad class since it contains simple synchronizing automata like those inČerný's series. Semisimplicity gives also the advantage of "factorizing" the problem of finding a synchronizing word into the sub-problems of finding "short" words that are zeros into the projection of the simple components in the WedderburnArtin decomposition. In the general case this last problem is related to the search of radical words of length at most (n − 1) 2 where n is the number of states of the automaton. We show that the solution of this "Radical Conjecture" would give an upper bound 2(n−1) 2 for the shortest reset word in a strongly connected synchronizing automaton. Finally, we use this approach to prove the Radical Conjecture in some particular cases andČerný's conjecture for the class of strongly semisimple synchronizing automata. These are automata whose sets of synchronizing words are cyclic ideals, or equivalently, ideal regular languages that are closed under taking roots.

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A Linear Bound on the K-Rendezvous Time for Primitive Sets of NZ Matrices

Azfar

Catalano

Charlier

et al. 2019

Developments in Language Theory

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A set of nonnegative matrices is called primitive if there exists a product of these matrices that is entrywise positive. Motivated by recent results relating synchronizing automata and primitive sets, we study the length of the shortest product of a primitive set having a column or a row with k positive entries (the k-RT). We prove that this value is at most linear w.r.t. the matrix size n for small k, while the problem is still open for synchronizing automata. We then report numerical results comparing our upper bound on the k-RT with heuristic approximation methods.Keywords: Primitive set of matrices, synchronizing automaton, Černý conjecture.Deciding whether a set is primitive is an NP-hard problem if the set contains at least three matrices (while it is still of unknown complexity for sets of two matrices) [3], and so is computing its exponent. For sets of matrices having at least a positive entry in every row and every column (called NZ [13] or allowable matrices [16,18]), the primitivity problem becomes decidable in polynomial-time [28], although computing the exponent remains NP-hard [13]. Methods for approximating the exponent have been proposed [6] as well as upper bounds that depend just on the matrix size; in particular, if we denote with exp N Z (n) the maximal exponent among all the primitive sets of n × n NZ matrices, it is known that exp N Z (n) ≤ (15617n 3 + 7500n 2 + 56250n − 78125)/46875 [3,33]. Better upper bounds have been found for some classes of primitive sets [13,17]. The NZ condition is often met in applications and in particular in the connection with synchronizing automata.

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Primitive digraphs with large exponents and slowly synchronizing automata

Cited by 50 publications

References 30 publications

Computing the shortest reset words of synchronizing automata

Computing the shortest reset words of synchronizing automata

Semisimple Synchronizing Automata and the Wedderburn-Artin Theory

A Linear Bound on the K-Rendezvous Time for Primitive Sets of NZ Matrices

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